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The world of advanced baseball statistics can be an intimidating place for those of us who slept our way through advanced algebra or haven't been a follower of the Bill James revolution from the beginning. 

Still, that doesn't mean that we should feel left out when it comes to another way of understanding and appreciating the game we all love. With that in mind, BLS stat doctor Alex Remington will explore a new advanced statistic each week during the offseason, providing a simple primer for the uninitiated.

Today's statistic: WAR

Good god, what does it stand for?: Wins Above Replacement. Say it again! Simply put, Wins Above Replacement means: how many wins did that player contribute to his team's win total above and beyond what they would have gotten from a "replacement value" player, someone they could have picked up off the scrap heap for next to nothing?

How they calculate WAR: WAR is probably the most popular total value stat out there today, a single number that attempts to quantify a player's worth by looking at his offense, defense (or pitching), defensive position, and the context of the year and league. (Bill James' Win Shares is  another total value stat, but it has been eclipsed in popularity by WAR.)

It's also one of the hardest to calculate, so hang with me here.

As Jabberwocky explains here on Beyond the Boxscore, WAR basically aggregates a lot of methods and insights from other advanced stats, like wOBA, FIP and UZR, all of which basically express a player's contributions to the team in terms of runs added or runs prevented. These run measures are then adjusted for that player's value "Above Replacement." A replacement player is defined as someone who is below average and should be easily obtainable, the sort of fringy cup-of-coffee guy you can find in AAA, on the waiver wire, or acquire for a PTBNL, a warm body who hurts the team the more he plays. With WAR anyone on the 25-man roster should have an implicit value just in keeping those fringy AAAA guys down on the farm and off the team.

After calculating the run measures, a positional adjustment is added to account for the relative importance of different positions and a league adjustment to account for the relative strength of the American League — which has a higher run environment thanks to the DH, higher salaries thanks to the presence of the Red Sox and Yankees and currently a greater concentration of talent — compared to the National League.

For a position player, you then add their wRAA, UZR, positional adjustment, and replacement adjustment to get their Runs Above Replacement. Then you scale all those contributions to be expressed in terms of total team wins. The usual scale is that 10 runs is equal to one team win. So WAR is equal to RAR divided by 10.

Believe it or not, it's much more complicated for a pitcher. In order to derive an equivalent of RAR from FIP, FIP must be rescaled to Runs Allowed rather than ERA (because unearned runs count for the purposes of RAR), then adjusted to the pitcher's innings pitched, and then the result must be adjusted for replacement level, for the run environment that the pitcher is creating, and for park effect. Dave Cameron wrote a seven-part series explaining the methodology and showed the formula in action here, but even he doesn't fully explain all the constants in his final formula. There's another way to derive it from Pitcher Winning Percentage, which is a variant on Bill James's Pythagorean Win Expectation formula. Devil_Fingers explains it here.

What WAR is good for: Absolutely nothing! Oh, excuse me. I've just been informed by the Committee on Obvious Humor not to make any more references to that song.

WAR is basically one attempt at a Grand Unified Stat, a single number that usefully expresses a player's worth and can be compared to all other players. Dave Cameron just wrote a fascinating piece for ESPN which used WAR analysis to argue that Chase Utley(notes) is actually a more valuable player than Albert Pujols(notes), because of the comparative value of their defensive positions and team contracts. Because WAR takes all of this into account, it's a great starting point for argument.

Also, as the Utley article illustrates, WAR is also an interesting way to look at player salaries. A full discussion on properly valuing player salaries is outside the scope of this article (and Colin Wyers did a great job here, here and here) but the basic assumption is this: there is a linear relationship between WAR and salary, and by multiplying a player's WAR by the average value of a win that year, you can come up with an approximate figure of how much money his performance that year was truly worth.

How WAR works: Conceptually, it's simple: WAR is a sum of the win value of a player's offense, defense, pitching, adjusted for that player's defensive position, playing time (thus keeping the replacement level players off the field) and year, park, and league context. The heavy lifting occurs in the individual calculations of the values and constants — as usual, I really just recommend you just use FanGraphs.

The fuzziest of all of these is the concept of the "replacement player." Tom Tango defines it as "the talent level for which you would pay the minimum salary on the open market, or for which you can obtain at minimal cost in a trade." On the other hand, we've all seen our teams struggle with players below replacement level, like Emilio Bonifacio(notes) and Yuniesky Betancourt(notes) (who had WAR of -0.4 and -2.1 last year, literally below replacement level). In the majors last year, there were eight players with a negative WAR. So "replacement level" is more of a theoretical conception rather than a concrete reality.

The positional adjustments are fixed: catchers, center fielders and shortstops are significantly more valuable than designated hitters and first basemen, so if you look at two players with similar offensive and defensive production at their positions, a catcher with those skills is worth about two more wins than a first baseman.

Generally speaking, the very biggest stars in the game, both pitchers and hitters, post a WAR of 7 or better. Ben Zobrist(notes) led all of baseball this year with a WAR of 8.6, which goes to show just how valuable his incredible versatility in the field was. Anything above 9 is extremely rare. (Since 2001, the first year that we have all of the data available for WAR calculations, only two players have ever posted a WAR above 10: Adrian Beltre(notes), the year he exploded to hit .334 and 48 homers with world-class defense at third, and Barry Bonds(notes), who did it every year from 2002-2004. Very good players are between 4 and 6 WAR; players between 1 and 3 are useful, and those with a WAR less than 1 are, by definition, easily replaceable.

When WAR doesn't work: An awful lot of fixed constants are used in the calculation of this stat, both in the individual run wOBA and FIP calculations and then in the positional, replacement, and league adjustments. Every model relies on assumptions, and WAR is no exception. But that simplicity comes at the price of the ability for the model to predict all the variance we see. Fortunately, a number of sabermetricians on the web have continued to go back and rejigger the constants every year, as more games are played and more data is added, so that the assumptions don't become outdated. Still, the caveat remains the same: there are a lot of different ways of calculating a player's worth, and a lot of different ways of choosing which on-field events to include in the model (intentional walks? double plays?) and which to ignore (temperature? wind?).

In addition, UZR is a useful defensive stat, but it's far from perfect; it frequently contradicts the findings of Plus/Minus, from John Dewan's Fielding Bible, which results from a video analysis of every play made by every defensive player, and there's no easy way to reconcile the contradiction. UZR can show some major fluctuations year to year — Nate McLouth(notes) had a UZR of -13.8 in 2008, then +3.6 in 2009, for example — which resulted in a nearly two-win swing in his defensive WAR. Ultimately, WAR is a terrific shorthand for a player's worth, but it's by no means the final word on a player.

Why we care about WAR: Two reasons. First, whatever its flaws, it's a stat that lets you express just about everything you can say about a player. (Hey, Yankee fans: Derek Jeter(notes) had a higher WAR than Hanley Ramirez(notes) last year, 7.6 to 7.4. It's true!) Second, because it lends itself to contract discussions, it's a great way for fans to visualize roster construction in terms of what each player contributes to the team. It's not the perfect stat — no single measure ever will be — but it's impressive in its scope, and awfully fun to use. Good God!

Next week's lesson: UZR

Previous lessons: BABIP, OPS+, FIP, wOBA, WPA

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